Apparatuses and Methods for Control and Self-Assembly of Particles into Adaptable Monolayers

ABSTRACT

Apparatuses and methods for the control and self-assembly of particles into adaptable monolayers and changing the relative position of a plurality of particles at an interface between two fluids, including applying an electric field perpendicular to the interface; moving the particles vertically in the interface in response to applying the electric field; moving the particles laterally within the interface in response to the electric field and capillary forces; maintaining the particles at the interface when moving the particles vertically; and maintaining the particles at the interface when moving the particles laterally.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from U.S. Provisional patentapplication Ser. No. 61/002,482, filed Nov. 9, 2007, and which isincorporated by reference herein

FIELD OF THE INVENTION

This invention relates to apparatuses and methods for control andself-assembly of particles into adaptable monolayers.

BACKGROUND OF THE INVENTION

Particles floating at an interface, in general, self-assemble or clusterbecause they deform the interface, thus giving rise to lateral capillaryforces which cause the particles to cluster. For example, cereal flakesfloating on the surface of milk cluster by this mechanism. Theattractive lateral capillary forces arise due to the fact that for twofloating particles, the deformed interface is such that the interfaceheight between the particles is lowered due to the interfacial tension.See, for example, M. A. Fortes, “Attraction and repulsion of floatingparticles”, Can. J. Chem. 60, 2889 (1982); W. A. Gifford and L. E.Striven, “On the attraction of floating particles”, Chem. Engrg. Sci.26, 287-297 (1971); Kralchevsky, P. A., V. N. Paunov, N. D. Denkov, I.B. Ivanov and K. Nagayama. “Energetical and force approaches to thecapillary interactions between particles attached to a liquid-fluidinterface”, J. Colloid and Interface Sci. 155, 420-437 (1993), J.Lucassen, “Capillary forces between solid particles in fluidinterfaces”, Colloids Surf. 65, 131-137 (1992); and P. Singh and D. D.Joseph, “Fluid dynamics of Floating particles”, J. Fluid Mech. 530,31-80 (2005). The lateral component of the capillary force acting on theparticles is attractive and causes them to move towards each other.

This naturally occurring phenomenon, however, produces monolayers thatdisplay many defects, lack order (both short and long ranged) and whosedistance between the particles cannot be controlled. These are threedrawbacks which seriously limit the range of applications one can targetusing this technique. In addition, such a phenomenon does not manifestitself on particles smaller than ˜10 μm, which further limits theapplications for this technique.

The vertical position of a floating particle within a two-fluidinterface is such that the total force acting on it in the directionnormal to the interface is zero, and this position determines the extentof interfacial deformation and lateral capillary forces. A particle thatis denser than the liquid below can float on its surface because thevertical component of capillary force, which arises due to thedeformation of the interface, balances the particle's buoyant weight.

As mentioned above, such a phenomenon does not manifest itself onparticles smaller than ˜10 μm. In particular, for a small particle ofradius a, the buoyant weight, which scales as a³, becomes negligiblecompared to the capillary force, which scales as a, and thereforeinsignificant interfacial deformation is needed for balancing thebuoyant weight of small particles. Consequently, the lateral capillaryforces due to the deformation of the interface are too small to movemicron and submicron sized particles, and thus, small particles, ingeneral, do not self-assemble under the action of capillary forcesalone. However, small particles can self-assemble if they are charged orif they have irregular contact lines.

Many envisioned applications of nanotechnology and fabrication ofmesoscopic objects strongly rely on the manufacturing of such micro- andnano-structured materials. Future progress in this area will criticallydepend upon our ability to accurately control the particles arrangement(e.g., lattice spacing, defect-free capability and long range order) inthree dimensions (3D) as well as in two dimensions (2D) for a broadrange of particle sizes, shapes, and types.

Accordingly, there is a need for improved methods and systems for thefabrication of self-assembled, defect-free adjustable monolayers ofparticles. Those and other advantages of the present invention will bedescribed in more detail hereinbelow.

SUMMARY OF THE INVENTION

The present invention provides methods for the self-assembly ofadaptable monolayers of particles. The present invention is applicableto a broad range of particle sizes and types, including nano-particlesand electrically neutral particles. Importantly, the methods of thepresent invention are capable of controlling the lattice spacingstatically or dynamically, forming virtually defect-free monolayers. Themethods utilize the application of an electric field perpendicular tothe interface, or having a component perpendicular to the interface, onwhich particles are floating which allows the manipulation of theclustering of particles floating at an interface. The method is capablenot only of expanding an already assembled monolayer, but also ofincreasing its level of order and of tuning the lattice spacing.

Such a control in 2D will lead to the manufacture of controlled particlemonolayers or ultra-fine porous membranes with adjustable, but regular(in the case of uniform particles and uniform electric field), poresizes, and therefore with adaptable mechanical, thermal, electricaland/or optical properties. Applications of such materials includes, butis not limited to controlled nanofluidic drug delivery patches withadjustable mass transfer properties across the patch; nanoporousmembranes and nanofilters, e.g., for separation of proteins based ontheir sizes; solid monolayers and fluid/fluid interfaces withdynamically adjustable properties (heat, mass, optical and electricalproperties); antireflection coatings (ARCs), e.g., to improve theefficiency of solar cells; photonic materials for optical circuits andelectronic nano-circuits with improved performance in absence ofdefects.

In one embodiment, the present invention is a method of changing therelative position of a plurality of particles at an interface betweentwo fluids. The method includes applying an electric fieldperpendicular, or having a component perpendicular, to the interface,moving the particles vertically in the interface in response to applyingthe electric field, moving the particles laterally within the interfacein response to the electric field and in response to capillary forces,maintaining the particles at the interface when moving the particlesvertically, and maintaining the particles at the interface when movingthe particles laterally.

In another embodiment, the present invention is a method of orienting aplurality of particles in a monolayer array, wherein the plurality ofparticles are located at an interface between two fluids. The methodincludes increasing (or decreasing) the intensity of an electric fieldperpendicular to the interface; increasing a distance between theplurality of particles in response to increasing (or decreasing) theelectric field perpendicular to the interface; maintaining the increaseddistance between the plurality of particles; allowing the plurality ofparticles to change their order and orientation after increasing thedistance between the plurality of particles; reducing (or increasing)the intensity of the electric field perpendicular to the interface; anddecreasing the distance between the plurality of particles in responseto decreasing (or increasing) the intensity of the electric fieldperpendicular to the interface.

In another embodiment the present invention is an apparatus including: acontainer; first and second electrodes located on opposite sides of thecontainer; a voltage source having an input, having an output connectedto the first electrode, and having an output connected to the secondelectrode; a sensor oriented to sense a characteristic of particles atan interface between two fluids in the container; a processor connectedto an output of the input device and connected to an input of thevoltage source; and a memory connected to the processor. The memoryincludes computer-readable instructions which, when executed by theprocessor, cause the processor to perform the steps of: determining acharacteristic of the particles at the interface via the input device;determining whether the characteristic sensed satisfies a predeterminedcondition; sending control signals to the voltage source to apply anelectric field perpendicular to the interface if the characteristic ofthe particles at the interface do not meet the predetermined condition;moving the particles vertically in the interface in response to theelectric field; moving the particles laterally within the interface inresponse to the electric field and in response to capillary forces; andmaintaining the particles at the interface when moving the particlesvertically and when moving the particles laterally.

Many variations are possible with the present invention, some of whichwill be described in the following detailed description of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will now be described, by way ofexample only, with reference to the accompanying drawings for thepurpose of illustrating the embodiments, and not for purposes oflimiting the invention, wherein:

FIG. 1 illustrates one embodiment of an apparatus according to thepresent invention.

FIG. 2 illustrates another embodiment of the apparatus according to thepresent invention.

FIG. 3 illustrates another embodiment of the apparatus according to thepresent invention.

FIGS. 4 and 5 illustrate embodiments of the method according to thepresent invention.

FIG. 6 illustrates another embodiment of the apparatus of the presentinvention.

FIG. 7 illustrates another embodiment of the apparatus according to thepresent invention.

FIG. 8 illustrates one embodiment of the operation of the presentinvention in the form of re-ordering particles into a more orderedarray.

FIG. 9 illustrates another embodiment of the operation of the presentinvention in the form of re-ordering particles into a more orderedarray.

FIG. 10 illustrates another embodiment of the operation of the presentinvention in the form of re-ordering particles into a more orderedarray.

FIG. 11 is a schematic of a heavier than liquid hydrophilic (wetting)sphere of radius a hanging on the contact line at θ_(c).

FIG. 12 is a schematic of one embodiment of an apparatus according tothe present invention.

FIG. 13 illustrates the electric field intensity on the domain midplanepassing through the particles centers for the device shown in FIG. 11(assuming that there are two particles on the interface), for variousdielectric constant values.

FIG. 14 illustrates electric field intensity on the domain midplane forthe device shown in FIG. 11 in which one particle is placed at theinterface.

FIG. 15 illustrates the vertical electrostatic force computednumerically and plotted as a function of the particle radius a.

FIG. 16 illustrates the vertical electrostatic force coefficient f_(v)plotted as a function of sin θ′_(c) for ε_(L)=1.1, 2, 5 and 50.

FIG. 17 illustrates the lateral dipole-dipole interaction force.

FIG. 18 illustrates the dipole interaction force coefficient f_(D).

FIG. 19 illustrates energies of capillary attraction (W_(c)) anddipole-dipole repulsion (W_(d)).

FIG. 20 illustrates the dimensionless equilibrium separation between twoparticles as a function of E for three values of the particle radius a.

FIG. 21 illustrates the equilibrium separation r_(eq)/(2 a) between twoparticles for a=37 and 53 μm as given by equation (12) and the actualmeasured values (denoted by Expt) are shown as functions of the voltageapplied.

FIG. 22 illustrates an embodiment of the present invention in which afilm is placed in between two fluids and the particles are locatedeither at the interface between the top fluid and the film, or at theinterface between the bottom fluid and the film, or at the twointerfaces.

FIG. 23 illustrates an embodiment of the present invention includingmore than one film between the fluids.

DETAILED DESCRIPTION OF THE INVENTION

The present invention applies generally to the controlled self-assemblyof millimeter to nano sized particles at a two-fluid interface using anelectric field. Under certain conditions, an externally applied electricfield can be used to control the spacing and movement of particlesfloating at a two-fluid interface. The present invention has manyapplications, such as to assemble a virtually defect free monolayerwhose lattice spacing can be adjusted by varying the electric fieldstrength. Although the invention works when both the electricalpermittivity and conductivity of the particles are non-zero, the physicsunderlying the present invention will be described in the case whereboth fluids and particles are perfect dielectrics and for this case the(capillary and electrical) forces acting on the particles will beanalyzed, an expression for the lattice spacing under equilibriumcondition will be deduced, and the dependence of the latter upon thevarious parameters of the system, including the particles' radius, thedielectric properties of the fluids and particles, the particles'position within the interface, the particles' buoyant weight and theapplied voltage will be studied. While for relatively large sizedparticles whose buoyant weight is much larger than the verticalelectrostatic force, the equilibrium distance increases with increasingelectric field, for submicron sized particles whose buoyant weight isnegligible, it decreases with increasing electric field. Forintermediate sized particles, the distance first increases and thendecreases with increasing electric field strength.

FIG. 1 illustrates an apparatus 10 according to one embodiment of thepresent invention. The apparatus 10 includes a container 12 and twoelectrodes 14.

The container 12 is used to hold first and second fluids 16, 18. Thefluids 16, 18 may have the same dielectric constant, or they may havedifferent dielectric constants. The fluids 16, 18 may be one liquid andone gas, or two liquids. The fluids 16, 18 form an interface 20 betweenthe fluids 16, 18, and the interface 20 may include particles 22. Thecontainer may have a bottom surface 26, a top surface 28, and sidesurfaces 30. The container 12 may also have shapes that do not includeall of those surfaces, such as an open top container 12, or a sphericalcontainer 12, or other shapes.

The fluids 16, 18 will generally be described as layers of fluids, withone layer 18 on top of another 16. As will be described in more detailherein, more than two fluids or fluid layers may be used with thepresent invention. In some embodiments, the term “film” is used for arelatively thin layer of fluid. However, the particular terms used inconnection with the fluids are not intended to limit the presentinvention, and many variations (and thickness) of the fluids arepossible with the present invention. These fluids 16, 18 (or fluidlayers) will generally be described in terms of the lower fluid 16having a depth that is greater than the diameter of the particles 22.However, in some embodiments of the present invention the lower fluid 16may be relatively shallow (e.g., less than the diameter of the particles22) so that the particles 22 (or some but not all of the particles 22 ifthe particles 22 have different characteristics) are in contact with thebottom surface 26 of the container 12. This embodiment may be used, forexample, to manipulate the particles 22 to produce films and coatings.Similarly, in other embodiments of the present invention, the upperfluid 18 may be relatively shallow (e.g., less than the diameter of theparticles 22) so that the particles (or some but not all of theparticles 22 if the particles 22 have different characteristics) are incontact with the top surface 28 of the container 12.

The interface 20 is formed between the fluids 16, 18. The interface 20may be formed by the two fluids 16, 18 contacting each other directly,or the interface 20 may include one or more additional fluids betweenthe fluids 16, 18, such as one or more films. For example, the interface20 may include a film whose thickness in the direction normal to theinterface 20 is smaller than the particles' 22 diameter. Other fluidfilms with different thicknesses may also be used with the presentinvention. The interface 20 will sometimes be described as being planar,which is true in a general sense as well as initially before theparticles are introduced on the interface 20. However, the interface 20is deformed around the particles 22 as described in more detail herein.In other embodiments, the electric field 24 may be used to deform theshape of the interface 20 so that it is curved or otherwise non-planar.

The particles 22 are located at the interface 20 between the fluids 16,18 and form a two-dimensional array of particles 22 at the interface 20.The particles 22 and the fluids 16, 18 are typically selected as thosewhich will move within the two dimensional plane of the interface 20 butresist moving out of the interface 20 and into the bulk fluids 16, 18.The particles may be of many different materials, shapes, and sizes. Forexample, the present invention will generally be described in terms ofparticles 22 being glass beads, although materials other than glass andshapes other than spheres may be used with the present invention.

The particles 22 will generally be described as being all the same,although this is not required for the present invention. For example,the particles 22 may have different sizes, different dielectricproperties, different wettabilities, different properties with regard tothe fluids 16, 18, and other variations. These differences between theparticles 22 can be used, for example, to provide for different particle22 spacing, different orientations and arrangements of the particles 22,and other variations according to the present invention. The particles22 will generally have homogeneous physical, electrostatic, optical,etc. properties either internally or on their surface, although this isnot required for the present invention. For example, the particles 22could have two or more faces (like the so-called Janus particles) orparticles with internal inhomogeneities like particles with a core, amembrane, etc.).

The electrodes 14 are used to apply an electric field 24 across theinterface 20 and affect the particles 22. In particular, the presentinvention can use the electric field 24 to change the vertical positionof the particles 22 relative to the interface 20. The electric field 24also gives rise to the repulsive electrostatic forces between theparticles 22. By doing so, the present invention can control the lateralcapillary and electrostatic forces acting on the particles and, thereby,control the lateral movement and lateral spacing of the particles 22 inthe interface 20.

The electrodes 14 will generally be described in terms of applying auniform electric field 24 across the interface 20. However, theelectrodes 14 may also be used to apply a non-uniform electric field 24across the interface 20. For example, the electric field 24 may increaseor decrease at different parts of the interface 20, and the variationsmay be gradual or they may be sudden. The creation of a non-uniformelectric field 24 may be accomplished, for example, through the use ofmore than two electrodes 14 or through the particular operation of theelectrodes 14. The non-uniform electric fields 24 can be used, forexample, to provide for non-uniform spacing of particles 22 at theinterface 20.

The electrodes 14 will generally be described in terms of applying anelectric field 24 perpendicular to the interface 20. This may beaccomplished, for example, by applying the electric field 24 exactlyperpendicular to the interface 20. However, a perfectly perpendicularelectric field 24 is not required with the present invention. On thecontrary, the present invention may apply the electric field 24 atorientations that are offset from perpendicular with respect to theinterface 20. In such cases, there will be a component of electric field24 that is perpendicular to the interface 20, and a component of theelectric field 24 that is parallel to the interface 20. As a result,applying an electric field 24 perpendicular to the interface 20 may beaccomplished with a perfectly perpendicular electric field 24 or with anelectric field 24 that is not perfectly perpendicular.

The present invention may use a parallel component of the electric field24 to control the particles 22. For example, if there is a parallelcomponent to the electric field 24, it may be used to affect the lateralmovement of the particles 22, so that the particles 22 may align alongthe parallel component of the electric field and thus form particlechains. In this case, the extent of the attractive forces between theparticles 22 and the possibility of particle chains formation willdepend on a number of factors, such as the strength of the parallelcomponent of the electric field 24 and the properties of the particles22. The perpendicular component of the electric field 24 is described inmore detail herein.

The present invention will generally be described in terms of two fluids16, 18, with one fluid located above the other and with a generallyplanar interface 20 between the fluids 16, 18. Terms such as “vertical”,“lateral”, “above”, “below”, “top”, “bottom”, and others are used in thecontext of that orientation. However, the present invention includesmany variations and embodiments, and the present invention is notlimited to the particular embodiments described herein.

FIG. 2 illustrates another embodiment of an apparatus 10 according tothe present invention. In that embodiment, the apparatus 10 has a squarecross-section and, for example, is used to assemble particles 22 on thesurface of corn oil 18. The distance between the electrodes 14 is 0.6 cmand the cross-sectional dimensions are 1.9×1.9 cm. The electric field 24is generated by two electrodes 14 that are mounted at the top and bottomsurfaces, and energized using AC voltage. The electrodes 14 may beinsulated by placing a layer of non-conducting material between thefluid 16 and the electrode 14, and between the fluid 18 and theelectrode 14. An AC electric field with a frequency of 100 Hz was usedto ensure that the influence of conductivity was negligible. Theelectrostatic force acts on particles in both vertical (F_(ev)) andlateral (F_(el)) directions. The particles 22 also experience capillaryforces which are not shown. The liquid 18 in the lower layer fillsapproximately one half of the container 12. The thickness of the lowerfluid 18 layer is adjustable and can be such that the particles 22 touchthe lower surface of the device.

The illustrated apparatus 10 was used to perform experiments withapproximately mono-disperse glass particles 22 of radius between 2 μmand 77.5 μm. The density and dielectric constant of glass particles is2.5 g/cm³ and 6.5, respectively, and those of corn oil 18 are 0.92 g/cm³and 2.87. The conductivity of corn oil 18 is 32.0 pSm⁻¹ and theinterfacial tension between air and corn oil is 33.5 dyne/cm. Anymoisture particles may contain was removed by drying particles in anoven for more than 3 hours at the temperature of 120° C.

FIG. 3 illustrates another embodiment of the present invention. In thatembodiment, the apparatus 10 includes a processor 32, memory 34, asensor 36, an output or display device 38, a voltage supply 40, and aninput device 42. The processor 32 is connected to the memory 34, thesensor 36, the output device 38, and the voltage supply 40. The memory34 includes computer readable instructions, such as computer hardware,software, firmware, or other forms of computer-readable instructionswhich, when executed by the processor 32, cause the processor 32 toperform certain functions, as described herein.

The processor 32 receives input from the sensor 36, and provides controlsignals to the voltage supply 40. The processor 12 also performs certainfunctions, as described herein.

The memory 34 can be any form of computer-readable memory, and may storeinformation in electrical form, magnetic form, optical form, or otherforms. The memory 34 includes computer readable instructions which, whenexecuted by the processor 32, cause the processor 32 to perform certainfunctions, as described herein. The memory 34 may be separate from theprocessor 32, or the memory 34 may be integrated with the processor 32.The memory 34 may also include more than one memory device, which may beintegrated with the processor 32, separate from the processor 32, orboth.

The sensor 36 may be a camera or other device for determining one ormore parameters of the particles 22. For example, the sensor 36 maymonitor the interface 20 at visible or invisible wavelengths and providesignals to the processor 32. The sensor 36 may be used, for example, todetermine the spacing between particles 22, whether particles are movingapart or together, or other characteristics.

The output device 38 may be a video display or other forms of outputtinginformation to a user.

The voltage source 40 receives control signals from the processor 32and, in response thereto, controls the electric field 24 by controllingthe voltage across the electrodes 14.

The input device 42 may be, for example, a keyboard, a touchscreen, acomputer mouse, or other forms of inputting information from a user tothe processor 32.

Many variations are possible with the system 10 according to the presentinvention. For example, more than one processor 32, memory 34, sensor36, output device 38, voltage supply 40, and input device 42 may bepresent in the system 10. In addition, devices not shown in FIG. 3 mayalso be included in the system 10, and devices shown in FIG. 3 may becombined or integrated together into a single device, or omitted. Forexample, in some embodiments the voltage supply 40 may be controlleddirectly by a user, such as through the input device 42, and theprocessor 32, memory 34, and sensor 36 may be eliminated.

In one embodiment, the sensor 36 monitors particles 22 at the interface20 and send signals to the processor 32 indicative of sensedcharacteristics of the particles 22 at the interface 20. The processor32 receives the signals from the sensor 36, determines whether theparticles 22 satisfy a desired condition, and sends to control signalsto the voltage source 40 to adjust the electric field 24 and change thecharacteristics of the particles 22 and to perform certain tasks asdescribed herein.

FIG. 4 illustrates one embodiment of a method according to the presentinvention. That method is directed generally to controlling the lateralmovement of particles 22 within the interface 20.

Step 60 includes applying an electric field 24 perpendicular to theinterface 20.

Step 62 includes moving the particles 22 vertically in response toapplying the electric field 24. As described in more detail herein, theapplication of the electric field 24 according to the present inventionwill, in general, cause the particles 22 to move vertically at theinterface 20.

Step 64 includes moving the particles 22 laterally within the interface20 in response to the electric field and capillary forces that act onthe particles 22 because of the deformation of the interface 20. Asdescribed in more detail herein, moving the particles 22 vertically willchange the interface 20 in the area around the particles 22, and thischange will induce lateral capillary forces acting on the particles 22,thereby causing the particles 22 to move laterally in the interface 20.Also as described in more detail herein, the application of the electricfield 24 polarizes the particles 22, which causes electrostaticparticle-particle interactions, thereby causing the particles 22 to moveapart.

Step 66 includes maintaining the particles 22 at the interface 20. Asdescribed in more detail herein, the particles 22 are caused to moveboth vertically (in general) and laterally according to the presentinvention, but the particles generally remain at the interface 20 andgenerally do not move into the bulk fluids 16, 18.

Many variations are possible with the present invention, and the methoddescribed herein may be varied and modified within the teachings of thepresent invention.

FIG. 5 illustrates another method according to the present invention.That method may be implemented, for example, with the apparatusillustrated in FIG. 3

Step 70 includes sensing a characteristic of the particles 22 at theinterface 20. The sensing 70 may be accomplished with, for example, thesensor 36.

Step 72 includes determining whether the characteristic of the particles22 that was sensed satisfies a predetermined condition. For example,with reference to FIG. 3, the processor 32 may process data from thesensor 36 and determine if the characteristics of the particles 22satisfy a predetermined condition. That condition may be, for example,the spacing between the particles 22, the order and arrangement of theparticles 22, the number of defects in the array of particles 22, orother conditions. The predetermined condition may, for example, bestored in the memory 34 and accessed by the processor 32.

Step 74 determines whether the characteristic sensed satisfies thepredetermined condition. If it does, then the method returns to step 70.If the characteristic sensed does not satisfy the predeterminedcondition, then the method proceeds to step 60 where the electric fieldis applied. In other words, if it is determined that the particles 22are not satisfactory without the application of the electric field, thensteps 60, 62, 64, and 66 are also performed as described above. If,however, it is determined that the particles 22 are satisfactory withoutthe application of the electric field, then the electric field is notapplied and steps 60, 62, 64, and 66 are not performed.

The method illustrated in this figure may be repeated many times, andthe characteristic of the particles that is sensed in step 70 may changeover time, the predetermined condition in step 72 may also change, andother parts of the method may change.

For example, the method may begin by sensing 70 the degree to which theparticles are ordered and/or the number of defects in the array ofparticles 22. If it is determined that the particles 22 are notsatisfactory, then the electric field may be applied in step 60 so thatthe particles 22 begin to move apart from each other.

The next time step 70 is performed, the characteristics of the particles22 that is sensed may be the distance between the particles 22, and theelectric field may be increased until that distance meets apredetermined condition or distance.

The next time step 70 is performed, the characteristic that is sensedmay again be the degree to which the particles 22 are ordered and/or thenumber of defects, and the electric field may be maintained until theparticles 22 rearrange themselves to meet the predetermined conditionwhile being spaced apart by the predetermined distance.

The next time step 70 is performed, the predetermined condition may bethe distance between the particles 22 and the electric field may bereduced until the particles have come together again.

The next time step 70 is performed, the characteristic sensed may be thedegree to which the particles are ordered and/or the number of defects.If the particles meet the predetermined condition then the method mayend or it may proceed to a different task. If the particles 22 stillfail to meet the predetermined condition, the process described abovemay be repeated or a modified method may be performed.

Many other variations of the present invention are also possible.

FIG. 6 illustrates another embodiment of the present invention in theform of a filter. In that embodiment, at least part of the container 12is transparent to one or more wavelengths of light or otherelectromagnetic energy of interest. The light or other electromagneticenergy (represented as “λ1+λ2”) enters the apparatus 10 and encountersthe particles 22 at the interface 20. The spacing of the particles 22can be adjusted by the processor 32 by controlling the voltage supply 40so that certain wavelengths of light or electromagnetic energy arepassed or reflected by the particles, as desired. By changing thespacing and/or arrangement of the particles 22, the wavelengths ofinterest can be changed and the filter according to the presentinvention can be made to be variable.

For example, the spacing of the particles 22 may be selected to reflectone or more wavelengths of light (λ1) and to pass one or morewavelengths of light (λ2). The wavelengths that are reflected and passedcan be changed by changing the spacing of the particles 22. Differentlight wavelengths may be handled by the superimposition of variousmonolayers. For instance, all the light may be absorbed in order toreduce light reflection (or even prevent reflection altogether) or, incontrast, all the light could be reflected. Many other variations arealso possible. As will be described in more detail herein, the presentinvention may utilize more than one interface 20, with particles 22 ateach of the more than one interface. In that way, different particle 22spacing and different particle 22 characteristics may be presentedsimultaneously to the light or other energy or mass passing through thecontainer 12.

The light or other electromagnetic energy may be carried to and from theapparatus 10 via optical fiber, free space device, waveguides, or byother means (collectively “50”). The present invention may also be usedfor applications other than light or electromagnetic filters.

FIGS. 7 a and 7 b illustrate another embodiment of the apparatus 10 inwhich the particles 22 are controlled to allow for the passage or theblockage of light or other energy passing through apparatus 10. Inparticular, particles 22 may be selected so that they absorb or reflectthe light or other energy of interest. Accordingly, the spacing of theparticles 22 can be controlled to selectively allow the light or otherenergy to pass through the apparatus 10 or to be blocked by theparticles 22.

In FIG. 7 a the particles 22 are controlled (as described in more detailherein) so that the particles 22 come together and form a barrier toblock the light or other energy.

In FIG. 7 b, the particles 22 are controlled (as described in moredetail herein) so that they are spaced apart, providing openings throughwhich the light or other energy may pass.

Many other variations are also possible with the present invention. Forexample, the present invention can be used to separate particle 22 fromeach other and then reassemble the particles 22 in a more ordered array,with fewer defects, more even spacing, and better short and long rangeorder. The present invention can also be used to form desired structuresin the particles 22 so as to form films and coatings from the particles22. As described above, the present invention can also be used to changeand control the spacing of particles 22, and thereby provide a methodand apparatus for various forms of (possibly nano-) porous membranes orfilters, such as light filters, particulate filters, and other types offilters. Those and other variations are possible with the presentinvention.

FIG. 8 illustrates one embodiment of the operation of the presentinvention in the form of re-ordering particles into a more orderedarray. In this embodiment, the particles 22 are glass, and the fluids16, 18 are air and oil. The average radius of the particles 22 is 40.5μm.

FIG. 8( a) illustrates the particles self-assembled under the action ofthe lateral capillary forces alone. The lattice is approximatelytriangular, but lacks long range order and contains many defects.

FIG. 8( b) illustrates the particles when a voltage V=5000 volt isapplied across the electrodes 14. The particles 22 move away from eachother and form a defect-free triangular lattice in which the distancebetween the particles is approximately 2.7 times the particle 22 radius.

FIG. 8( c) illustrates the particles 22 after the applied voltage isslowly decreased to 0 volt. The particles 22 touch each other in a wellorganized triangular lattice and the lattice exhibits long range order.Notice that the number of lattice defects is considerably reducedcompared to that in (a) where the monolayer was assembled under theaction of capillary forces alone.

FIG. 8( d) illustrates the particles 22 in a monolayer when 3500 voltare applied across the electrodes 14. The lattice spacing is smallerthan in (b).

The electric field 24 is applied perpendicular to the interface 20 whichallows the manipulation of the clustering of particles 22 floating atthe interface 20 such that the particles 22 form well-controlled, andactive monolayers. The technique is capable of not only expanding analready assembled monolayer but also tuning the lattice spacing. FIGS. 8b and 8 c show that a cluster of glass particles 22 at the interface 20can be expanded so that the particles 22 in the dilated state arearranged on a triangular (also called hexagonal) lattice, and shrunkagain by decreasing the electric field 24 intensity. It is interestingto point out that the particle 22 arrangements are very different beforeturning on the electric field 24 (FIG. 8 a) and after turning on theelectric field 24 and decreasing it to zero (FIG. 8 c). Irregularitiesin spacing of the particles 22 are present in absence of an electricfield 24 (FIG. 8 a). However, when the electric field 24 is turned onand slowly decreased to zero (FIGS. 8 b and 8 c), the particles 22 gaina well-ordered, triangular lattice arrangement and maintain it as thelattice distance decreases until the particles 22 touch each other.Therefore, the holes in between the particles 22 (i.e., the holes of theultra-fine porous membrane) in this new state are also regular—the onlyirregularities remaining being due to the variation in the size of theparticles 22 themselves.

FIG. 9 illustrates assembled glass particles 22 in the form of amonolayer for particles with an average radius of 12 μm. FIG. 9 issimilar to that illustrated in FIG. 8, except that the particles 22 usedin FIG. 9 are smaller.

FIG. 9 a illustrates the monolayer assembled under the action ofcapillary forces alone. FIG. 9 b illustrates the monolayer formed byturning on the electric field 24, allowing the particles 22 to reorderthemselves, and then decreasing the electric field 24 to zero. Themonolayer in this figure is relatively less organized than that in FIG.8 because the size variation of the particles 22 in this figure issignificantly greater than that of FIG. 8.

FIG. 10 illustrates another example of the assembly of particles 22according to the present invention. In that example, glass particles 22are floating at an air-oil interface. The average radius of theparticles is 23.5 μm. (a) Particles 22 self-assemble under the action ofthe lateral capillary forces alone. The lattice is approximatelytriangular, but lacks long range order and contains many defects. (b)When a voltage V=5000 volt is applied, particles 22 move away from eachother and form a defect-free triangular lattice in which the distancebetween the particles 22 is approximately 2.7 times the particle radius.This order is maintained as the electric field is either increased ordecreased, particularly as it is decreased to zero.

1. Analysis Overview

A more detailed analysis of the present invention will now be presented.The present invention includes the application of an electric field inthe direction normal to a two-fluid interface to control the spacing ofparticle located at the interface. The present invention may be used,for example, as a process of particles self-assembly at the interface.The two-fluid interface can be between a liquid and a gas or aninterface between two liquids.

The present invention allows for the clustering of particles atinterfaces to be controlled, which is important because it allows formodifying the interfacial properties of two-phase systems and alsobecause it can be used, for example, for the self-assembly of particlesto form monolayers at two-liquid interfaces. See, for example, N.Bowden, I. S. Choi, B. A. Grzybowski, G. M. Whitesides, “Mesoscaleself-assembly of hexagonal plates using lateral capillary forces:synthesis using the ‘capillary bond’, J. Am. Chem. Soc. 121, 5373-5391(1999); N. A. Bowden, A. Terfort, J. Carbeck, G. M. Whitesides,“Self-assembly of mesoscale objects into ordered two-dimensionalarrays”, Science 276, 233-235 (1997); and B. A. Grzybowski, N. Bowden,F. Arias, H. Yang, G. M. Whitesides, “Modeling of menisci and capillaryforces from the millimeter to the micrometer size range”, J. Phys. Chem.B 105, 404-412 (2001). See also the references in those documents.

FIG. 11 is a schematic of a heavier than liquid hydrophilic (wetting)sphere 22 of radius a hanging on the contact line at θ_(c). The point ofextension of the flat meniscus on the sphere 22 determines the angle θ₁and h₂ is defined as h₂=a(cos θ_(c)−cos θ₁). The angle α is fixed by theYoung-Dupré law and θ_(c) by the force balance.

In equilibrium, the vertical position of a floating particle 22 within atwo-fluid 16, 18 interface 20 is such that the sum of the forces actingon the particle 22 in the direction normal to the interface 20 is zero.A particle 22 denser than the liquid 16 below can float on its surfacebecause the vertical component of the capillary force, which arises dueto the deformation of the interface 20, balances the particle's 22buoyant weight. For a small particle 22 of radius a, the buoyant weight,which scales as a³, becomes negligible, and therefore only a smallinterfacial 20 deformation is needed in this case for the verticalcapillary force to balance the buoyant weight. Consequently, the lateralcapillary forces due to this small deformation of the interface are toosmall to move micron and nano sized particles, and thus, smallparticles, in general, do not self-assemble. It is known that forparticles floating on the air-water interface 20 the attractivecapillary forces are significant only when the particle 22 radius islarger than ˜10 μm. See, for example, N. Bowden, I. S. Choi, B. A.Grzybowski, G. M. Whitesides, “Mesoscale self-assembly of hexagonalplates using lateral capillary forces: synthesis using the ‘capillarybond’”, J. Am. Chem. Soc. 121, 5373-5391 (1999); N. A. Bowden, A.Terfort, J. Carbeck, G. M. Whitesides, “Self-assembly of mesoscaleobjects into ordered two-dimensional arrays”, Science 276, 233-235(1997); and B. A. Grzybowski, N. Bowden, F. Arias, H. Yang, G. M.Whitesides, “Modeling of menisci and capillary forces from themillimeter to the micrometer size range”, J. Phys. Chem. B 105, 404-412(2001).

This restriction, however, does not apply to particles trapped in thinfilms with a thickness smaller than the particle diameter. In fact,particles ranging from protein macromolecules to millimeter sizedparticles can self-assemble in such thin films. See, for example, P. A.Kralchevsky and K. Nagayama, “Capillary interactions between particlesbound to interfaces, liquid films and biomembranes”, Advances in Colloidand Interface Science 85, 145-192 (2000). Moreover, small particles canself-assemble if they are charged or if they have irregular contactlines. See, for example, D. Y. C. Chan, J. D. Henry Jr. and L. R. White,“The interaction of colloidal particles collected at the fluidinterface”, J. Colloid Interface Sci. 79, 410 (1981); and D. Stamou andC. Duschl, “Long-range attraction between colloidal spheres at theair-water interface: The consequence of an irregular meniscus”, PhysicalRev. E 62, 5263-5272 (2000). Furthermore, the two-fluid interface of thepresent invention includes a two-fluid interface with a thin film ofanother liquid at the interface, wherein the thickness of the film isless than the diameter of the particles at the interface.

Experiments with the present invention show that particles floating at atwo-fluid interface can be self-assembled to form monolayers by applyingan electric field normal to the interface, and that the lattice spacingof the monolayer thus formed can be adjusted by varying the electricfield strength. The technique also leads to the formation of virtuallydefect free monolayers with long range order and, in principle, can beused for manipulating the assembly of sub micron sized particles intwo-fluid interfaces. It thus overcomes all the shortcomings of theusual capillarity induced clustering mentioned earlier.

In the context of fluids and particles which are perfect dielectrics, wenext discuss the dependence of the electrostatic force acting on aparticle upon the parameters of the system such as the dielectricconstants of the fluids and particles, the particles' position withinthe interface, and the distance between the particles. It is also shownthat the component of the electrostatic force normal to the interfacealters the deformation of the interface due to the particles, and thusthe magnitude of the lateral capillary forces. The lateral component ofthe electrostatic force acting on the particles is repulsive, and itsbalance with the attractive capillary force is used to determine theequilibrium distance between the particles. It is assumed that both theparticles and the fluids considered here are perfect dielectrics. Ifthey are not (they are conductive), the dielectric constant needs to bereplaced by the real part of the complex permittivity (with includesboth the dielectric constant and the conductivity) and double layereffects in the vicinity of the particle surface may not be negligible.

2. Electrostatic Forces, Governing Equations and DimensionlessParameters

FIG. 12 is a schematic of one embodiment of an apparatus 10 according tothe present invention. The apparatus was used to assemble particles 22on the surface 20 of corn oil 16. The distance between neighboringparticles 22 is controlled by adjusting the magnitude of the appliedvoltage. An AC electric field with frequency 100 Hz was used to make theinfluence of conductivity negligible.

In the experiments described herein, the electric field away from theinterface 20 is uniform and normal to the undeformed interface 20. Whilean isolated uncharged dielectric particle placed in a uniform electricfield becomes polarized, it does not experience any electrostatic force.This, however, is not the case for a particle 22 floating at a two-fluidinterface 20 between two fluids 16, 18 with different dielectricconstants because of the mismatch between the dielectric constants ofthe two fluids involved. Moreover, from symmetry the electrostatic forceacting on an isolated spherical particle at an interface can only be inthe direction normal to the interface, but depending on the parametervalues it can be either upward or downward. If the particle is charged,a coulomb force also acts on the particle. In addition, when there areother particles present at the interface they interact with each othervia electrostatic (to first order, dipole-dipole) interactions. We nowproceed to compute the electrostatic force acting on the particles atthe interface, first briefly describing the numerical method used.

2.1 Numerical Method

Let us denote the domain containing the two fluids and sphericalparticles (of identical radii and properties) by Ω, the interior of theith particle and its surface by P_(i)(t) and ∂P_(i)(t), respectively,and the domain boundary by Γ. To calculate the electric field E, wefirst solve the electric potential problem for φ in Ω, namely ∇(ε∇φ)=0subjected to the boundary conditions on the particle surfaces and thetwo-fluid interface. On the particle surface ∂P_(i)(t), the conditionsread φ₁=φ₂,

${{ɛ_{c}\frac{\partial\varphi_{1}}{\partial n}} = {ɛ_{p}\frac{\partial\varphi_{2}}{\partial n}}},$

where φ₁ and φ₂ are the electric potentials in the liquid and particle,and ε_(c) and ε_(p) are the dielectric constants of the fluid andparticle. A similar boundary condition is applied at the two-fluidinterface. The electric potential is prescribed on the electrodes asconstant values and the normal derivative of the potential is taken tobe zero on the remaining domain boundary. The electric field is thendeduced from the equation E=∇φ. The Maxwell stress tensor σ_(M) is givenby

${\sigma_{M} = {{ɛ\; E\; E} - {\frac{1}{2}{ɛ\left( {E \cdot E} \right)}I}}},$

where I is the identity tensor and the electrostatic force acting on theith particle is then obtained by integrating σ_(M) over its surface,i.e.

F_(DEP) = ∫_(∂P_(i)(t))σ_(M) ⋅ ns,

where n is the unit outer normal on the surface of the ith particle. Thecomputational domain in our finite element code is discretized using atetrahedral mesh and the boundary conditions are imposed on the surfaceof the particles. The resulting linear system of equations is solvedusing a multigrid preconditioned conjugate gradient method. See, forexample, J. Kadaksham, P. Singh and N. Aubry, “Dynamics of pressuredriven flows of electrorheological suspensions subjected to spatiallynon-uniform electric fields”, Journal of fluids engineering 126, 170-179(2004); J. Kadaksham, P. Singh and N. Aubry, “Dielectrophoresis of nanoparticles”, Electrophoresis 25, 3625-3632 (2004); J. Kadaksham, P. Singhand N. Aubry, “Dielectrophoresis induced clustering regimes of viableyeast cells”, Electrophoresis 26, 3738-3744 (2005); J. Kadaksham, P.Singh and N. Aubry, Manipulation of Particles Using Dielectrophoresis,Mechanics Research Communications 33, 108-122 (2006); N. Aubry and P.Singh, “Control of Electrostatic Particle-Particle Interactions inDielectrophoresis”, Euro Physics Letters 74, 623-629 (2006).

2.2 Vertical Electrostatic Force

As noted above, even though the applied electric field away from theinterface is uniform, a particle within the interface experiences anelectrostatic force normal to the interface due to a jump in dielectricconstants across the interface. If the interface does not contain anyparticles, the electric field is normal to the interface and itsintensity in the lower and upper fluids is constant, while changingdiscontinuously at the interface according to the boundary conditionstated above.

FIG. 13 illustrates the electric field intensity on the domain midplanepassing through the particles centers for the device shown in FIG. 12,for various dielectric constant values. The spherical particles ofradius a are placed so that their centers are aligned with thenon-deformed interface and at a distance 2.6 a of each other. Thespheres alter the electric field distribution, and experience anelectrostatic force. The dielectric constants of the upper fluid, lowerfluid and the particles were set to: (a) ε_(a)=1, ε_(L)=5 and ε_(p)=2.The vertical force is 4.16 and the lateral force is −0.156; (b) ε_(a)=1,ε_(L)=2 and ε_(p)=5. The vertical force is 3.39 and the lateral force is−0.34; (c) ε_(a)=1, ε_(L)=0.5 and ε_(p)=2. The vertical force is −1.49and the lateral force is −0.124. In all cases, the lateral electrostaticforce is repulsive while the vertical electrostatic force can be eitherupward (a, b) or downward (c).

In the case of two spherical particles placed at the interface, theelectric field distribution on the domain mid plane, which passesthrough the spheres centers, is shown in FIG. 13 for three differentcombinations of dielectric constants when the particle centers are atthe undeformed interface. To compute the electrostatic force acting onthe particles, we must first determine the shape of the interface which,in general, is deformed due to the presence of the particles. Thedeformed interface shape can be computed by solving the equations ofmotion for the fluids and the particles and for the interface, subjectedto the contact angle and boundary conditions. This, however, isdifficult to do analytically and is not necessary to use in the presentinvention. The dielectric constant of the upper fluid is assumed to beone, while those of the lower fluid and the particles are varied. Thefigure shows how the particles presence at the interface modifies theelectric field distribution. In particular, the electric field is theweakest in the fluid or particle region in which the dielectric constantis the largest and the strongest in the region for which the dielectricconstant is the smallest. For example, in FIG. 13 a where the dielectricconstant of the particles has a value in between the dielectricconstants of the lower and upper fluids, the electric field intensity isthe strongest in the upper fluid, the weakest in the lower fluid and inbetween these two values inside the particles. It follows that themagnitude of the vertical electrostatic force on the particle, as wellas its direction, depends on the dielectric constant values. Forexample, the force is positive (acts against gravity) in FIGS. 13 a and13 b, and negative in FIG. 13 c. The lateral electrostatic force isrepulsive in all three cases, but its magnitude depends on thedielectric constant values and is different for the three cases.

We now turn to the electric field distribution around a particle whenits position within the interface is altered. FIG. 14 illustrateselectric field intensity on the domain midplane for the device shown inFIG. 12 in which one particle is placed at the interface. The dependenceof the electric field upon the position of the particle within theinterface is studied. The dielectric constants were set to: ε_(a)=1,ε_(L)=5 and ε_(p)=0.5. The particle alters the electric fielddistribution and experiences an electrostatic force in the verticaldirection. The direction of the force, as well as its magnitude, dependson the particle position within the interface. (a) The sphere center isat the interface. The vertical electrostatic force is 0.354 (in theupward direction). (b) The sphere center is at a distance of 0.6 a belowthe interface. The vertical electrostatic force is −2.22 (in thedownward direction). (c) The sphere center is at a distance of 0.6 aabove the interface. The vertical electrostatic force is 0.553.

Because of the presence of the interface the electric field around theparticles is not symmetric and, as a result, the particle experiences anelectrostatic force in the direction normal to the interface. For thesecalculations, the dielectric constants of the upper fluid, the lowerfluid and the particle are set to ε_(a)=1.0, ε_(L)=5.0, ε_(p)=0.5. FIG.14 a shows that a particle with its center at the undeformed interfaceexperiences an electrostatic force in the upward direction. The electricfield distributions for the cases when the particle center is below andabove the undeformed, flat interface are shown in FIGS. 14 b and 14 c.The electrostatic force is in the upward direction in the former caseand in the downward direction in the latter case.

Our numerical results show that the vertical component of theelectrostatic force in a DC field (or time averaged force in an ACfield) acting on a particle can be written as:

$\begin{matrix}{F_{ev} = {a^{2}ɛ_{0}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)}E^{2}{{f_{v}\left( {\frac{ɛ_{L}}{ɛ_{a}},\frac{ɛ_{p}}{ɛ_{a}},\theta_{c},\frac{h_{2}}{a}} \right)}.}}} & (1)\end{matrix}$

Here a is the particle radius, E=V₀/L is the average electric fieldstrength away from the particle (or the RMS value of the electric fieldin an AC field), ε_(p), ε_(a) and εL are the dielectric constants of theparticle, the upper fluid and the lower fluid, respectively, andε₀=8.8542×10⁻¹² F/m is the permittivity of free space. Here, L is thedistance between the electrodes, V₀ is the voltage difference applied tothe electrodes, and

$f_{v}\left( {\frac{ɛ_{L}}{ɛ_{a}},\frac{ɛ_{p}}{ɛ_{a}},\theta_{c},\frac{h_{2}}{a}} \right)$

is a dimensionless function of the included arguments (θ_(c) and h₂being defined in FIG. 11). The dependence of the force on the particleradius a is quadratic which was established numerically as shown in FIG.15. In particular, FIG. 15 illustrates the vertical electrostatic forcecomputed numerically and plotted as a function of the particle radius a,along with the best power law fit, showing the quadratic dependence ofthe force on the particle radius. The dielectric constants were assumedto be: ε_(a)=1, ε_(L)=2 and ε_(p)=1.5. The factor

$\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)$

ensures the fact that the force is zero when

$\frac{ɛ_{L}}{ɛ_{a}} = 1$

as the fluids dielectric constants are the same in this case. It isassumed that the interface is flat and intersects the particle's surfaceat θ_(c) (see FIG. 11). One of the focuses of this patent is on thebehavior of small floating particles for which the interfacialdeformation is negligible, and thus θ_(c)≈π−α where α is the contactangle (see FIG. 11). In addition, the influence of electrowetting isconsidered only in the sense that if there is a change in the effectivecontact angle, the resulting change in the particle's position withinthe interface can be accounted for by changing θ_(c). See, for example,F. Mugele and J. Baret, “Electrowetting: from basics to applications”,J. Phys.: Condens. Matter 17, R705-R774 (2005). Also notice that thedependence of the electrostatic force on the particle radius a isquadratic compared to the cubic dependence of the dielectrophoreticforce which acts on a particle in a non-uniform electric field.

FIG. 16 illustrates the vertical electrostatic force coefficient f_(v)plotted as a function of sin θ′_(c) for ε_(L)=1.1, 2, 5 and 50. Thedielectric constant of the particle is 2.0 (a) and 0.5 (b) and that ofthe upper liquid is 1.0. In FIG. 16, we have set ε_(a)=1, and ε_(p)=2.0(FIG. 16 a) or 0.5 (FIG. 16 b), and studied the force coefficient f_(v)as a function of sin θ′_(c) for various values of ε_(L), where

$\theta_{c}^{\prime} = {\theta_{c} - {\frac{\pi}{2}.}}$

Notice that it is sufficient to consider the case where ε_(L)>ε_(a)because the electric force for the corresponding case where ε_(L)<ε_(a)can be deduced from the results shown in FIG. 16 by simply reversing thedirection of the force. The figure shows that for sin θ_(c)<0 the forcecoefficient f_(v) is positive for all cases investigated here and itsmagnitude decreases with increasing ε_(L) . A positive value of theelectrostatic force for θ′_(c)<0 implies that the particle is pushedinto the upper liquid whose dielectric constant is smaller. However,from FIG. 16 a we also notice that for ε_(L)=5 and sin θ′_(c)>0.8, andfor ε_(L)=50 and sin θ′_(c)>0.3, f_(v) is negative, implying that theparticle in these two cases is pushed into the lower liquid whosedielectric constant is larger. In other words, if the particle center islocated below the interface at a distance larger than a criticaldistance (whose value depends on ε_(L)), the particle is pushed furtherdownwards; otherwise, the electrostatic force pushes the particleupwards. Therefore, in the presence of an electric field the interfaceacts like a barrier because it opposes the motion of the particlesacross the interface. The figure, however, also suggests, since thevertical force is largely positive, that the electric force pushesparticles into the fluid whose dielectric constant is smaller providedthe latter are able to cross the interface barrier. The figure alsoshows that for ε_(L)=1.1 the force is maximal when θ′_(c)=0 and that theangle θ′_(c) for which the force is maximal decreases with increasingε_(L).

Similarly, for ε_(p)=0.5, shown in FIG. 16 b, there is a critical valueof θ′_(c) at which f_(v) changes sign, and the critical value of θ′_(c)decreases with increasing ε_(L). This, as in FIG. 16 a, implies that theelectrostatic force pushes the particle away from the flat interface,and thus, as it was the case above, the interface acts as a barrier forthe particles. The critical value of θ′_(c) in FIG. 16 b, however, issmaller than in FIG. 16 a at the same ε_(L) value. The actual verticalposition (θ′_(c)) of a particle is, of course, determined by the balanceof the buoyant weight, the vertical component of the capillary force,and the vertical component of the electrostatic force.

2.3 Lateral Electrostatic Forces

The dipole-dipole interaction force between two dielectric spheresimmersed in a fluid with the dielectric constant ε_(a) and subjected toa uniform electric field, in the point-dipole limit, is given by thefollowing well-known expression in spherical coordinates:

$\begin{matrix}{{F_{D}\left( {r,\theta} \right)} = {{f_{0}\left( \frac{a}{r} \right)}^{4}\left( {{\left( {{3\cos^{2}\theta} - 1} \right)e_{r}} + {\sin \; 2\theta \; e_{\theta}}} \right)}} & (2)\end{matrix}$

where f₀=12πε₀ε_(a)a²β²E² (E being the magnitude of the uniform electricfield along the z-axis), θ denotes the angle between the z-axis and thevector r joining the centers of the two particles, r=|r|,

$\beta = \frac{ɛ_{p} - ɛ_{a}}{ɛ_{p} + {2ɛ_{a}}}$

is the Clausius-Mossotti factor, and ε_(p) is the dielectric constant ofthe particle. See, for example, H. A. Pohl, 1978 Dielectrophoresis(Cambridge: Cambridge University Press); Jones, T. B. Electromechanicsof particles 1995, Cambridge University Press; F. Mugele and J. Baret,“Electrowetting: from basics to applications”, J. Phys.: Condens. Matter17, R705-R774 (2005); D. J. Klingenberg, S. Van Swol, C. F. Zukoski,“Simulation of electrorheological suspensions”, J. Chem. Phys. 91, pp.7888-7895 (1989); and W. R. Smythe, “Static and Dynamic Electricity”,3rd ed., McGraw-Hill, New York (1968). For particles trapped at theinterface, the electric field is perpendicular to the line joining thecenters of the particles, i.e., θ=π/2, and thus the interaction force isrepulsive and tangential to the interface.

However, the above expression is not applicable to particles floating inan interface between two fluids with different dielectric constants, asthe fluid's dielectric constant changes discontinuously across theinterface. The computations described above were used to show that thelateral interaction force can be written as

$\begin{matrix}{{F_{D}(r)} = {ɛ_{0}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} + 1} \right)}a^{2}{E^{2}\left( \frac{a}{r} \right)}^{4}{f_{D}\left( {\frac{ɛ_{L}}{ɛ_{a}},\frac{ɛ_{p}}{ɛ_{a}},\theta_{c},\frac{h_{2}}{a}} \right)}}} & (3)\end{matrix}$

where f_(D) is a dimensionless function of the included arguments, withthe force depending upon the sixth power of the particle radius a and onthe fourth power of the inverse of the distance between the particles asshown in FIG. 17 a-b.

FIG. 17 illustrates the lateral dipole-dipole interaction force computednumerically is plotted as a function of (a) the dimensionless distancer/a between the particles and (b) the particle radius a for a fixeddistance r between the particles. The best power law fits, showing thatthe force depends on the inverse of the fourth power of the distancebetween the particles and on the sixth power of the particle radius, arealso shown. The dielectric constants are assumed to be ε_(a)=1, ε_(L)=2and ε_(p)=1.5.

As was the case for (1), the above expression is obtained by assumingthat the interface is flat and that it intersects the sphere's surfaceat θ_(c). The force also depends on the dielectric constants of the twofluids involved, and the positions θ_(c) of the particles within theinterface. The latter in this study is assumed to be the same for thetwo particles. However, if particles were not of the same type or size,their positions θ_(c) within the interface would be different, and theinteraction force would be even more complex. The experimentaltechnique, however, would still be applicable with a possibly varyinggap in between particles of various types.

FIG. 18 illustrates the dipole interaction force coefficient f_(D)plotted as a function of sin θ′_(c) for ε_(L)=1.1, 2, 5 and 50. Thedielectric constant of the particle is 2.0 (a) and 0.5 (b) and that ofthe upper liquid is 1.0. The distance between the particles is 2.6 a.

FIG. 18 displays the force coefficient f_(D) as a function of sin θ′_(c)for ε_(a)=1, and ε_(p)=2.0 (FIGS. 18 a) and 0.5 (FIG. 18 b). The forceis repulsive for all values of ε_(L) investigated. From FIG. 18 a wenote that for ε_(p)=2.0 and ε_(L)=50, the magnitude of f_(D) is maximumwhen θ′_(c) is around zero, but is relatively independent of forε_(L)=1.1 and 5.0. Also notice that for ε_(L)=2.0, as expected, f_(D)goes to zero when the sphere is completely submerged in the lower liquidas, in this case, the dielectric constant of the particles is the sameas that of the lower fluid.

For the case corresponding to ε_(p)=0.5 shown in FIG. 18 b, f_(D)increases in magnitude with increasing θ′_(c) when θ′_(c)<0 forε_(L)=1.1 and 2.0. This is due to the fact that the dielectric constantof the lower fluid is larger than that of the upper fluid. However, forε_(L)=5 and 50, f_(D) attains a maximum value in magnitude and thendecreases with increasing θ′_(c). This result suggests that for largervalues of ε_(L) the interface enhances the repulsive force between theparticles.

The repulsive interaction energy W_(D) between two particles can beobtained by integrating (3) with respect to r, which gives

$\begin{matrix}{{W_{D}(r)} = {{- \frac{1}{3}}ɛ_{0}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} + 1} \right)}a^{2}{E^{2}\left( \frac{a^{4}}{r^{3}} \right)}{f_{D}\left( {\frac{ɛ_{L}}{ɛ_{a}},\frac{ɛ_{p}}{ɛ_{a}},\theta_{c},\frac{h_{2}}{a}} \right)}}} & (4)\end{matrix}$

Let us assume that ε_(a)=2.0, ε_(L)=4.0, E=3×10⁶ volt/m, f_(D)=3.1, andr=2 a. For these parameter values, the interaction energy is shown as afunction of the particle radius in FIG. 19. For a=1 μm,W_(D)(r)=˜1.67×10⁴ kT and for a=100 nm, W_(D)(r)=˜16.7 kT, where k isthe Boltzman constant and T is the temperature, indicating that therepulsive electrostatic force is larger than the random Brownian forceacting on the particles. This shows that the electrostatic repulsiveforce (3) can be used to manipulate nanoparticles within a two-fluidinterface.

2.4 Vertical Force Balance in Equilibrium

We next consider the vertical force balance for a spherical particlefloating within the interface between two immiscible fluids. The buoyantweight F_(b) of the particle is balanced by the capillary force F_(c)and the electrostatic force F_(ev), that is

F _(e) +F _(ev) +F _(b)=0.   (5)

The buoyant weight is given by

${F_{b} = {{- g}\; \rho_{L}a^{3}{f_{b}\left( {\frac{\rho_{a}}{\rho_{L}},\frac{\rho_{p}}{\rho_{L}},\theta_{c},\frac{h_{2}}{a}} \right)}}},$

where g is the acceleration due to gravity, ρ_(p) is the particledensity, ρ_(a) and ρ_(L) are the densities of the upper and lowerfluids, θ_(c) and h₂ are defined in FIG. 11, and f_(b) is a function of

$\frac{\rho_{a}}{\rho_{L}},\frac{\rho_{p}}{\rho_{L}},{\theta_{c}\mspace{14mu} {and}\mspace{14mu} {\frac{h_{2}}{a}.}}$

It is easy to deduce from FIG. 11 that the capillary force F_(c) takesthe expression F_(c)=−2π γ a sin θ_(c) sin(θ_(c)+α), where α is thecontact angle. Therefore, equation (5) can be rewritten as

$\begin{matrix}{F_{c} = {{{- 2}{\pi\gamma}\; a\; \sin \; \theta_{c}{\sin \left( {\theta_{c} + \alpha} \right)}} = {{g\; \rho_{L}a^{3}{f_{b}\left( {\frac{\rho_{a}}{\rho_{L}},\frac{\rho_{p}}{\rho_{L}},\theta_{c},\frac{h_{2}}{a}} \right)}} - {a^{2}ɛ_{0}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)}E^{2}{f_{v}\left( {\frac{ɛ_{a}}{ɛ_{L}},\frac{ɛ_{p}}{ɛ_{L}},\theta_{c},\frac{h_{2}}{a}} \right)}}}}} & (6)\end{matrix}$

In dimensionless form, the previous equation reads

$\begin{matrix}{{2\pi \; \sin \; \theta_{c}{\sin \left( {\theta_{c} + \alpha} \right)}} = {{{- B}\; {f_{b}\left( {\frac{\rho_{a}}{\rho_{L}},\frac{\rho_{p}}{\rho_{L}},\theta_{c},\frac{h_{2}}{a}} \right)}} + {{W_{E}\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)}{{f_{v}\left( {\frac{ɛ_{a}}{ɛ_{L}},\frac{ɛ_{p}}{ɛ_{L}},\theta_{c},\frac{h_{2}}{a}} \right)}.}}}} & \left( 6^{\prime} \right)\end{matrix}$

Here B=ρ_(L)a²g/γ is the Bond number and

$W_{E} = {ɛ_{0}ɛ_{a}\frac{a\; E^{2}}{\gamma}}$

is the electric Weber number.

As the particle radius a approaches zero, the Bond numberB=ρ_(L)a²g/γ→0. In this limit, in the absence of an electrostatic force,the right hand side of equation (6′) is zero and thus sin (α+θ_(c))≈0 orθ_(C)¢π−α (see FIG. 11). This means that a small particle floats so thatthe interfacial deformation is insignificant. Hence, the lateralcapillary force, which arises from the interfacial deformation, in thislimit, is also insignificant. As noted earlier, for particles floatingon water, this limit is reached when the particles radius isapproximately 10 μm. See, for example, M. A. Fortes, “Attraction andrepulsion of floating particles”, Can. J. Chem. 60, 2889 (1982); W. A.Gifford and L. E. Scriven. “On the attraction of floating particles”,Chem. Engrg. Sci. 26, 287-297 (1971); Kralchevsky, P. A., V. N. Paunov,N. D. Denkov, I. B. Ivanov and K. Nagayama, “Energetical and forceapproaches to the capillary interactions between particles attached to aliquid-fluid interface”, J. Colloid and Interface Sci. 155, 420-437(1993); J. Lucassen, “Capillary forces between solid particles in fluidinterfaces”, Colloids Surf. 65, 131-137 (1992); P. Singh and D. D.Joseph, Fluid dynamics of Floating particles, J. Fluid Mech. 530, 31-80(2005).

Another important limit is the case for which the Bond number approacheszero, but W_(E) does not. This situation arises, for instance, for smallparticles when the magnitude of the electric field is sufficientlylarge. The equilibrium position of a particle within the interface inthis case is determined by the balance of the interfacial andelectrostatic forces alone. The interface is then deformed by theparticle, and so the lateral (electric field induced) capillary forcesare present and can cause particles within the interface to cluster.

2.5 Interfacial Deformation and Lateral Capillary Force

In equilibrium, the external vertical force acting on a particle isbalanced by the vertical component of the capillary force which, asnoted earlier, arises because of the deformation of the interface. Theprofile of the deformed interface around a particle can be obtained byintegrating Laplace's equation and using the boundary conditions that(i) the interface far away from the particle is flat and (ii) the anglebetween the interface and the horizontal at the particle surface isknown in terms of the total external force acting on the particle. Itcan be shown that the interface height η(r) at a distance r from aspherical particle is given by:

η(r)=a sin(θ_(c)+α)K ₀(qr)   (7)

where K₀(qr) is the modified Bessel function of zeroth order and

$q = {\sqrt{\frac{\left( {\rho_{L} - \rho_{a}} \right)g}{\gamma}}.}$

In obtaining the above expression we have ignored the influence of theelectrostatic stress on the interface, including the stress that arisesdue to the presence of the particle, and assumed that the interfacialdeformation is small. For more on the interface height η(r) at adistance r from a spherical particle, see M. A. Fortes, “Attraction andrepulsion of floating particles”, Can. J. Chem. 60, 2889 (1982); W. A.Gifford and L. E. Scriven. “On the attraction of floating particles”,Chem. Engrg. Sci. 26, 287-297 (1971); Kralchevsky, P. A., V. N. Paunov,N. D. Denkov, I. B. Ivanov and K. Nagayama, “Energetical and forceapproaches to the capillary interactions between particles attached to aliquid-fluid interface”, J. Colloid and Interface Sci. 155, 420-437(1993); J. Lucassen, “Capillary forces between solid particles in fluidinterfaces”, Colloids Surf 65, 131-137 (1992); P. Singh and D. D.Joseph, Fluid dynamics of Floating particles, J. Fluid Mech. 530, 31-80(2005); P. A. Kralchevsky and K. Nagayama, “Capillary interactionsbetween particles bound to interfaces, liquid films and biomembranes”,Advances in Colloid and Interface Science 85, 145-192 (2000); and M. M.Nicolson, “The interaction between floating particles”, Proc. CambridgePhilosophical Soc., 45, 288 (1949).

Let us consider a second particle at a distance r from the firstparticle. The height of the second particle is lowered because of theinterfacial deformation caused by the first particle, and thus the workdone by the electrostatic force and gravity (buoyant weight) is

W _(c)=−η(r)(F _(ev) +F _(b)).   (8)

Notice that the electrostatic force is due to a field that is externalto the fluid-particle system, as is the gravitational field, andtherefore the work done by both fields is treated in a similar manner.In this analysis, we will ignore the work done by the electrostaticstress that acts on the two-fluid interface. In addition, this analysisof the behavior of two particles does not account for the multi bodyinteractions (which could be accounted for by summing the forces exertedby all other particles on one given particle). Using equations (6) and(7), equation (8) can be rewritten as

$\begin{matrix}\begin{matrix}{W_{c} = {{- \frac{\left( {F_{ev} + F_{b}} \right)^{2}}{2{\pi\gamma}}}{K_{0}({qr})}}} \\{= {{- \left( {{{- ɛ_{0}}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)}a^{2}E^{2}f_{v}} + {\frac{4}{3}\pi \; a^{3}\rho_{p}g\; f_{b}}} \right)^{2}}\frac{1}{2{\pi\gamma}}{K_{0}({qr})}}}\end{matrix} & (9)\end{matrix}$

FIG. 19 illustrates energies of capillary attraction (W_(c)) anddipole-dipole repulsion (W_(d)), in kT units, are plotted against theparticle radius. For W/(kT)>1, the capillary attraction and thedipole-dipole repulsion are stronger than the Brownian force for allparticles sizes down to a radius of approximately 100 nm. The parametersare ε_(a)=2.0, ε_(L)=4.0, E=3×10⁶ volt/m, f_(v)=1, f_(D)=1, γ=0.01,ρ_(a)=1 kg/m³, pρ_(L)=1000, ρ_(p)=3000 kg/m³ and r=2 a.

In FIG. 19, the interaction energy W_(c) due to the lateral capillaryforce is plotted as a function of the particle radius. The parametervalues are ε_(a)=2.0, ε_(L)=4.0, E=3×10⁶ volt/m, f_(v)=1, γ=0.01,ρ_(a)=1 kg/m³, ρ_(L)=1000, ρ_(p)=3000 kg/m³ and r=2 a. The figure showsthat for these parameter values, the interaction energy (9) issignificant for nano sized particles.

The lateral capillary force between two particles is therefore given by

$\begin{matrix}\begin{matrix}{F_{lc} = {- \frac{W_{c}}{r}}} \\{= {\left( {{{- ɛ_{0}}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)}a^{2}E^{2}f_{v}} + {\frac{4}{3}\pi \; a^{3}\rho_{p}g\; f_{b}}} \right)^{2}\frac{q\; {K_{1}({qr})}}{2{\pi\gamma}}}}\end{matrix} & (10)\end{matrix}$

where K₁(qr) is the modified Bessel function of first order. When thetwo particles are far away from each other, the above reduces to

$\begin{matrix}{F_{lc} = {{- \left( {{{- ɛ_{0}}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)}a^{2}E^{2}f_{v}} + {\frac{4}{3}\pi \; a^{3}\rho_{p}g\; f_{b}}} \right)^{2}}\frac{1}{2{\pi\gamma}\; r}}} & (11)\end{matrix}$

Notice that the lateral capillary force depends on the net verticalforce acting on the particle, which includes its buoyant weight and thevertical electrostatic force. The force varies as the fourth power ofthe applied electric field, and if the electrostatic force and thebuoyant weight are in the same direction, the electric field enhancesthe lateral capillary forces among the particles.

However, it is noteworthy that the vertical electrostatic force may notbe in the same direction as the buoyant weight, and if this is the casethere is a critical value of the electric field strength for which thenet vertical force acting on the particle is zero. The lateral capillaryforce among the particles under these conditions would also be zero;this suggests that the electric field can be used to decrease, or eveneliminate, capillarity induced attraction among the particles. If theelectric field strength is increased further, the particles move upwardin the interface and the capillary forces arise again but the interfacenear the particles would be curved downwards. Here we wish to note thatthe capillary force can cause particles to interact with each other onlywhen the associated interaction energy is greater than kT, and thereforewhen the net external vertical force acting on the particles is smallthe latter are not likely to cluster as their motion would be governedby thermal fluctuations.

2.6 Spacing Between Particles

The dimensionless equilibrium separation r_(eq)/(2 a) between twoparticles can be obtained by equating the repulsive electrostatic force(3) and the above attractive capillary force (11). After simplification,we obtain

$\begin{matrix}{\frac{r_{eq}}{2a} = {\frac{1}{2}\left( \frac{2{\pi ɛ}_{0}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} + 1} \right)}\gamma \; E^{2}f_{D}}{{a\left( {{{- ɛ_{0}}{ɛ_{a}\left( {\frac{ɛ_{L}}{ɛ_{a}} - 1} \right)}E^{2}f_{v}} + {\frac{4}{3}\pi \; a\; \rho_{p}g\; f_{b}}} \right)}^{2}} \right)^{\frac{1}{3}}}} & (12)\end{matrix}$

This expression gives the dependence of r_(eq)/(2 a) on the parametersof the problem. However, we remind the reader that the dimensionlessparameters f_(v), f_(D) and f_(b) themselves depend on severalparameters (which is not reproduced in equation (12) for the sake ofsimplicity). Notice that r_(eq)/(2 a) decreases with increasing particleradius a.

FIG. 20 illustrates the dimensionless equilibrium separation between twoparticles plotted as a function of E for three values of the particleradius a. (a) a=10⁻³ m, r_(eq)/(2 a) increases with increasing E. (b)a=1 μm, r_(eq)/(2 a) decreases with increasing E. (c) The buoyant weightand the vertical electrostatic force are in the same direction anda=4×10⁻⁵ m. For E small, r_(eq)/(2 a) increases with increasing E, butfor E large it decreases with increasing E. (d) The buoyant weight andthe vertical electrostatic force are in the opposite directions anda=4×10⁻⁵ m. For small values of E, r_(eq)/(2 a) increases withincreasing E. There is a critical value of E for which the lateralcapillary force is zero and thus particles only experience the repulsiveelectrostatic force and r_(eq)/(2 a) approaches infinity. For E large,r_(eq)/(2 a) decreases with increasing E.

We now consider two limiting cases of the previous expression. The firstis the case of relatively large particles for which the buoyant weightis much larger than the vertical electrostatic force. In this situation,equation (12) implies that r_(eq)/(2 a) increases with increasingelectric field strength as E^(2/3). This is approximately the case fora=10⁻³ m, as shown in FIG. 20 a. These conclusions are in agreement withthe experimental data reported in N. Aubry, P. Singh, M. Janjua & S.Nudurupati, “Micro- and nano-particles self-assembly for defect-free,adjustable monolayers”, Proceedings of the National Academy of Sciences,105, 3711-3714 (2008) for particles with r=˜10⁻³ m. The attractivecapillary forces for such particles primarily originate in theinterfacial deformation due to their buoyant weight, and the repulsiveforce is due to the dipole-dipole interaction between them.

The second limiting case is that of relatively small sized particles forwhich the buoyant weight is negligible compared to the verticalelectrostatic force. From equation (12), after the buoyant weight isneglected, we obtain that r_(eq)/(2 a) decreases with increasingelectric field strength as E_(−2/3). As shown in FIG. 20 b, thislimiting case is approximately reached for a=1 μm. Both attractive andrepulsive forces in this case are due to the applied electric field(since the buoyant weight is negligible). The attractive part varies asthe fourth power of the electric field, but is long ranged (varies asr⁻¹). The repulsive part, on the other hand, varies as the square of theelectric field, but is short ranged (varies as r⁻⁴).

Here we wish to distinguish the above result with the model presented inM. G. Nikolaides, A. R. Bausch, M. F. Hsu, A. D. Dinsmore, M. P.Brenner, C. Gay and D. A. Weiz. “Electric filed induced capillaryattraction between like-charged particles at liquid interfaces”, Nature420, 299-301 (2002), to explain the observation that small chargedparticles within an interface between water and a nonpolar liquid, suchas air or oil, form periodic arrangements. They showed that in additionto interacting electrostatically with each other, the charged particlesexperience lateral capillary forces that arise because of thedeformation of the interface (also see L. Foret and A. Wurger. “Electricfield induced capillary interaction of charged particles in polarinterfaces”, Phys. Rev. Lett. 92, 058302-1 (2004); R. Aveyard, B. P.Binks, J. H. Clint, P. D. I. Fletcher, T. S. Horozov, B. Neumann, V. N.Paunov, J. Annesley, S. W. Botchway, D. Nees, A. W. Parker, A. D. Wardand A. N. Burgess. “Measurement of long-range repulsive forces betweencharged particles at an oil-water interface”, Phys. Rev. Lett. 88,246102-1 (2002); and K. D. Danov and P. A. Kralchevky, “Electric forcesinduced by a charged colloid particle attached to the water-nonpolarfluid interface”, Journal of Colloid and Interfaces Science 298, 213-231(2006)). The interfacial deformation in their experiments was due to avertical electrostatic force that acts on the particle within theinterface because of its charge and not due to the particle's weight(which is negligible). No external electric field was applied in theirexperiments, whereas for the invention described herein an externalelectric field is applied.

In the intermediate range between the two previous limiting cases, theattractive capillary force is a result of the net vertical force actingon the particles, which includes both the buoyant weight and thevertical electrostatic force. In an actual physical system, thisintermediate range corresponds to particle radii between ˜10 and ˜100μm. We now consider first the case in which the electrostatic force andthe buoyant weight are in the same direction. The latter dominates whenthe electric field strength is small, and therefore, the distancebetween the particles increases with increasing electric field (see FIG.20 c). However, as the electric field strength is increased to a levelwhere the vertical electrostatic force is much larger than the buoyantweight, the distance between the particles decreases with increasingelectric field strength. Here, it is important to note that this resultcan be observed in experiments only if the electrostatic and capillaryforces are larger than the Brownian forces. Furthermore, the formerforces cannot be observed if they do not cause an observabledeterministic motion. This may be the case if the distance between theparticles is too large.

We next consider the case in which the electrostatic force and thebuoyant weight are not in the same direction. In this case, there is acritical value of the electric field strength at which the lateralcapillary force is zero. This corresponds to the situation where the sumof the buoyant weight and the vertical electrostatic force on theparticle is zero. As a result, as shown in FIG. 20 d, r_(eq)/(2 a)approaches infinity because the only lateral force the particlesexperience is the repulsive electrostatic force. However, since therepulsive electrostatic force decays as the fourth power of the distancebetween the particles, in experiments, the particles are expected tomove only to a distance at which the associated interaction energybecomes comparable to kT. Another interesting feature of the curve inFIG. 20 d is that a further increase of the electric field strengthcauses the lateral capillary forces to increase (since the sum of thebuoyant weight and the vertical electrostatic force on the particle isagain non zero) and r_(eq)/(2 a) to decrease.

3.0 Discussion and Conclusions

In view of explaining in detail the clustering of particles, we havestudied the electrostatic and capillary forces acting on a particlewithin a two-fluid interface in the presence of both an externallyapplied electric field and other particles. Specifically, we havedetermined the dependence of the electrostatic force upon the dielectricproperties of the fluids and the particles, as well as the position ofthe particle within the interface. It was assumed that the particles andthe two fluids involved are perfect dielectrics, and that the particlesare spherical. These assumptions, however, are not necessary for thetechnique to work. The electrostatic force was found to containcomponents both normal and tangential to the interface. The formerarises because the dielectric constants of the two fluids involved aredifferent (and is thus zero for two fluids of identical dielectricconstants) and the latter is due to the dipole-dipole interactions amongthe particles. The component of the electrostatic force normal to theinterface is shown to vary as the square of the particle radius, a². Forsufficiently large distances between the particles, the lateralelectrostatic force between two particles varies as a⁶ and decreaseswith increasing distance between the particles as r⁻⁴. We have alsoshown that when E˜3×10⁶ volt/m, the electrostatic forces can be used tomanipulate the distance between nano sized particles floating at atwo-fluid interface. Expressions of the various forces involved, as wellas the equilibrium distance between the particles were given in N.Aubry, P. Singh, M. Janjua & S. Nudurupati, “Micro- and nano-particlesself-assembly for defect-free, adjustable monolayers”, Proceedings ofthe National Academy of Sciences, 105, 3711-3714 (2008).

The normal component of the electrostatic force, including its sign,depends on the dielectric constants of the fluids and particles. Theequilibrium particle position θ_(c) of a particle within the interfaceis determined by the balance of the buoyant weight, the verticalinterfacial force and the vertical electrostatic force. For smallspherical particles, in the absence of an electric field the particle'sposition is primarily determined by the contact angle since the buoyantweight is negligible. Our numerical results show that when thedielectric constant of the upper fluid is smaller than that of the lowerfluid and the particle's center is above the undeformed interface (thisis the case for a small particle which is non-wetting with the lowerliquid), the electrostatic force is in the upward direction. If, on theother hand, the particle center is below the undeformed interface (thisis the case for a small particle which wets the lower liquid), there isa critical value of θ_(c) at which the electrostatic force changesdirection. The critical value of θ_(c) depends on the dielectricconstants of the fluids and the particle. Therefore, in the presence ofan externally applied electric field, the interface acts like a barrierto the particles: the electrostatic force pushes the particles below theinterface downwards and those above the interface upwards. The overalltendency of the electric force, however, is to push particles into thefluid region whose dielectric constant is smaller, but this can occuronly if the particles have sufficient energy to cross the electricinterface barrier.

In equilibrium, the net vertical force acting on a particle at theinterface, which includes the electrostatic force and the buoyantweight, is balanced by the vertical capillary force which arises becauseof the deformation of the interface. The deformation of the interface,in turn, gives rise to lateral capillary forces which cause particles atthe interface to cluster. More specifically, it is shown that themagnitude of these lateral forces is determined by the square of the netvertical force acting on the particle which includes both the buoyantweight and the vertical electrostatic force. The lateral capillaryforces are long ranged and depend on the fourth power of the electricfield intensity.

The buoyant weight and the vertical electrostatic force, however, maynot be in the same direction, and when this is the case the electricfield, in fact, reduces lateral capillary forces. If the electrostaticforce and buoyant weight are in the same direction, the electric fieldenhances lateral capillary forces. This is an important result,especially for micron and sub micron sized particles for which thebuoyant weight is negligible, because it shows that the clusteringbehavior of particles, including that of small particles, can becontrolled using an externally applied electric field.

The equilibrium distance between two particles was obtained by equatingthe attractive capillary and repulsive electrostatic forces. Equilibriumis possible because the attractive capillary force between the particlesis long ranged (decays as r⁻¹) and dominates the electrostatic repulsiveforce which is short ranged (decays as r⁻⁴) when the distance betweenthe particles is large. The opposite is true when the distance betweenthe particles is small. The equilibrium distance was shown to depend onthe particle radius, the electric field intensity, the buoyant weight,the particle's position within the interface and the dielectricconstants. These results are in agreement with the recent experimentsreported in N. Aubry, P. Singh, M. Janjua & S. Nudurupati, “Micro- andnano-particles self-assembly for defect-free, adjustable monolayers”,Proceedings of the National Academy of Sciences, 105, 3711-3714 (2008),which show that the equilibrium distance between particles can becontrolled by adjusting the electric field strength (see FIG. 10).

FIG. 21 illustrates the equilibrium separation r_(eq)/(2 a) between twoparticles for a=37 and 53 μm as given by equation (12) and the actualmeasured values (denoted by Expt) are shown as functions of the voltageapplied to the device described in FIG. 12. The electric forcecoefficients were numerically estimated to be f_(v)=0.27, f_(D)=0.019,and f_(b)=0.64. From the experimental photographs, we estimatedθ_(c)=76.5 degrees for the particles with a=37 μm, and this value wasused for both cases. The agreement between the theory and theexperimental data is very good, especially when the distance between theparticles is more than 2.5 a, considering that there are no adjustableparameters.

The theoretical results presented here correctly capture the trendsobserved in experiments. For example, the variation of the dimensionlessequilibrium distance r_(eq)/(2 a) between two particles with theelectric field strength, and also with the particle radius is predictedcorrectly as shown in FIG. 21. The figure displays r_(eq)/(2 a) given byequation (12), along with the actual measured values, as functions ofthe voltage applied to the device described in N. Aubry, P. Singh, M.Janjua & S. Nudurupati, “Micro- and nano-particles self-assembly fordefect-free, adjustable monolayers”, Proceedings of the National Academyof Sciences, 105, 3711-3714 (2008). The data is presented for a=37 and53 μm. The distance between two particles increases with increasingelectric field and with decreasing particle radius. From equation (12)we know that in the limiting cases the dimensionless distance betweentwo particles varies as E^(β). For relatively large sized particles(a>˜1000 μm), for which the buoyant weight dominates, β=2/3, and forsubmicron sized particles, β=˜2/3. For the data presented in FIG. 21,the distance between the particles increases with increasing E.According to equation (12), for submicron sized particles the distancebetween two particles should decrease with increasing electric fieldstrength. At present, such experimental data for micron and submicronsized particles is not available, and therefore we are unable to verifythe predictions of our theory for this size range. This reversal in theparticle separation with increasing electric field strength, as notedbefore, is a consequence of the fact that the attractive capillary forceis not a result of the particles' buoyant weight, but instead arisesfrom the vertical electric force acting on the particles. To get smallparticles to cluster one may thus have to increase the electric fieldintensity.

Although the present invention has been described in terms of specificembodiments, many variations are possible.

FIG. 22 illustrates an embodiment of the present invention in which theinterface 20 between the fluids 16, 18 includes a film of a third fluid80. In that embodiment, the film 80 has a thickness that is smaller thanthe diameter of some of the particles 22. However, in other embodiments,the film 80 may be thicker or comparable to the diameter of theparticles 22. In some embodiments, the particles 22 are homogeneous andhave the same characteristics, and in other embodiments, they are nothomogeneous and/or have different characteristics. In such embodiments,for example, some of the particles 22 may have a diameter greater thanthe thickness of the film 80 and other particles 22 may have a diameterthe same as or less than the thickness of the film 80. Similarly, allparticles 22 may not float at the same height relative to the film 80,so that some particles 22 may be entirely within the film 80, someparticles 22 may extend below the film 80, and some particles 22 mayextend above the film 80.

In the illustrated embodiment 22, the third fluid 80 is placed betweenthe fluids 16, 18. The illustrated embodiment has an interface betweenfluids 16 and 80, and an interface between fluids 18, and 80. It ispossible with the illustrated embodiment to operate as a two interfacesystem with particles 22 at both interfaces (e.g., particles at theinterface between fluids 16 and 80, and particles 22 at the interfacebetween fluids 18 and 80). The manner in which the system operates willdepend on factors such as the thickness of the fluid 80, and thecharacteristics of the particles 22 and the fluids 16, 18, 80.

For example, if the thickness of the fluid 80 is small compared to thediameter of the particles 22, then the system may be more convenientlyviewed as having one interface (e.g., the film 80 is sufficiently thinand the particles sufficiently large that the interface between fluids16 and 80 and the interface between fluids 18 and 80 are treated as onlyone interface). In that case, for example, large particles 22 floatingat the interface between fluids 16 and 80 will interfere with themovement of large particles 22 floating at the interface of fluids 80and 18. Also, large particles 22 will float so that they are both at theinterface between fluids 16 and 80, and at the interface between fluids18 and 80. Thus, under these conditions particles 22 will experiencecapillary forces from both of these interfaces. However, if thethickness of the fluid 80 is large compared to the diameter of theparticles 22, then the system may be more conveniently viewed as havingtwo interfaces (one between fluids 16 and 80, and a second betweenfluids 80 and 18). In that case, for example, small particles 22floating at one interface may move freely within their interface withoutinterfering with particles 22 floating and moving at the otherinterface. As used herein, “interface” can mean the place where twofluids meet, or “interface” can mean two or more interfaces treated as asingle interface, such as a thin fluid or film 80 between two otherfluids 16, 18.

Although the fluid 80 between the first and second fluids 16, 18 isoften referred to as a “film”, the use of the term “film” is notintended to limit the thickness of the fluid 80. Furthermore, althoughthe third fluid/film 80 will generally be illustrated as having athickness significantly less than that of the first and second fluids16, 18, the third fluid 80 is not limited to having a thickness that issignificantly less than that of the first and second fluids 16, 18.

FIG. 23 illustrates another embodiment of the present inventionincluding two fluid layers or films 80, 82 between the first and secondfluids 16, 18. In other embodiments, the present invention may includeadditional fluids or films between the first and second fluids 16, 18.This system may be considered to have two interfaces 20 formed by thetwo additional fluids or films 80, 82. However, the system may also beviewed as having three interfaces where the fluids meet (e.g., between16 and 18, between 80 and 82, and between 82 and 18). The particles 22may be located at one or more than one of the various interfaces. Forexample, particles 22 may be located at each of the interfaces betweenthe fluids 16, 18, 80, 82, or at only some of the interfaces. In thiscase, capillary forces involving the multiple interfaces will act on theparticles located at such interfaces.

Many variations are possible with the present invention. For example, inother embodiments, some of the interfaces 20 may not include particles22. In addition, each of the fluids 16, 18, 80, 82 may be a differentmaterial, or the same material may be used for more than one fluid layer(such as in the case where two or more fluid materials are alternated toform the various layers of fluids 16, 18, 80, 82). Similarly, all of thefluids 16, 18, 80, 82 may have the same thickness, or they may havedifferent thicknesses. For example, in the illustrated embodiment, theupper film 80 may have a greater thickness than the lower film 82,although this is not required in the present invention. As describedabove, the particles 22 may be homogeneous or they may have differentcharacteristics, and the characteristics of the particles may be thesame in each of the different films 20, or the characteristics of theparticles 22 in one film may be different from those in other films 20.Other variations are possible.

Although the present invention has been described in terms of specificembodiments and implementations, the present invention is applicable toother methods, apparatuses, systems, and technologies. The examplesprovided herein are illustrative and not limiting, and other variationsand modifications of the present invention are contemplated. Othervariations and modifications of the present invention are possible andcontemplated, and it is intended that the foregoing specification andthe following claims cover such modifications and variations.

1-24. (canceled)
 25. An apparatus, comprising: a container; first andsecond electrodes located on opposite sides of the container; a voltagesource having an input, having an output connected to the firstelectrode, and having an output connected to the second electrode; asensor oriented to sense a characteristic of particles at an interfacebetween first and second different fluids in the container; a processorconnected to an output of the input device and connected to an input ofthe voltage source; a memory connected to the processor and includingcomputer-readable instructions which, when executed by the processor,cause the processor to perform the steps of: determining acharacteristic of the particles at the interface via the input device;determining whether the characteristic sensed satisfies a predeterminedcondition; sending control signals to the voltage source to apply anelectric field perpendicular to the interface if the characteristic ofthe particles at the interface do not meet the predetermined condition;moving the particles vertically in the interface in response to theelectric field; moving the particles laterally within the interface inresponse to the electric field and capillary forces; and maintaining theparticles at the interface when moving the particles vertically and whenmoving the particles laterally.
 26. The apparatus of claim 25, whereinthe particles deform the interface around the particles, and wherein thecapillary forces arise because of the deformation of the interface. 27.The apparatus of claim 25, wherein the first and second fluids havedifferent dielectric constants.
 28. The apparatus of claim 25, whereinthe first and second fluids have the same dielectric constants.
 29. Theapparatus of claim 25, wherein the interface is a film of a third fluidbetween the first and second fluids.
 30. The apparatus of claim 29wherein: the particles have a diameter; and the third fluid has athickness less than the diameter of the particles.
 31. The apparatus ofclaim 25, wherein: the container has a bottom surface; the first fluidis on the bottom surface of the container; and the first fluid has athickness; and the particles have a diameter that is greater than thethickness of the first fluid.
 32. The apparatus of claim 31, wherein theparticles contact the bottom surface of the container.
 33. The apparatusof claim 25, wherein: the container has a bottom surface; the firstfluid is on the bottom surface of the container; the second fluid isabove the first fluid, and the second fluid has a thickness, and theparticles have a diameter that is greater than the thickness of thesecond fluid.
 34. The apparatus of claim 33, wherein: the container hasa top surface; and the particles contact the top surface of thecontainer.